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Building Polygons
In the last two investigations, you explored the relationship between the
number of sides of a polygon and the measure of its interior angles. Now
you will turn your attention to the sides of a polygon.
How do the side lengths of a polygon affect its shape?
You can use polystrips and fasteners like these:
to build polygons with given side lengths and study their properties.
70
Shapes and Designs
TEKS! lAKS
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6{l1jC Select all appropriate problemosoiving strategy inducting looking fora pattern.
6(13}A Make conjectures from patterns.
Bridges, towers, and other structures contain many triangles in their design.
Why are triangles used so frequently in construction?
Building Triangles
Make a triangle using the steps below. Sketch and label your results.
Step 1 Roll three number cubes and record the sum. Do this two more
times, so that you have three sums.
Step 2 Using polystrips, try to make a triangle with the three sums as side
lengths. If you can build one triangle, try to build a different
triangle with the same side lengths.
Repeat Steps 1 and 2 to make several triangles.
A. 1. List each set of side lengths that did make a triangle.
2. List each set of side lengths that did not make a triangle.
3. What pattern do you see in each set that explains why some sets of
numbers make a triangle and some do not?
4. Use your pattern to find two new sets of side lengths that will make
a triangle. Then find two new sets of side lengths that will not make
a triangle.
B. Can you make two different triangles from the same three side lengths?
c. Why do you think triangles are so useful in construction?
Homework starts on page 76.
Investigation 4 Building Polygons
71
TEKS! TAKS 6(11)( Select an appropriate problem-solving strategy induding looking for a pattern.
G(U)A Make conjectures from patterns.
_
.
You need four side lengths to make a quadrilateral.
Will any four side lengths work?
Can you make more than one quadrilateral from four side lengths?
Building Quadrilaterals
A. 1. Use polystrips to build quadrilaterals with each of the following
sets of numbers as side lengths. Try to build two or more
different quadrilaterals using the same set of side lengths.
6,10,15,15
3,5,10,20
8,8,10,10
12,20,6,9
Sketch and label your results to share with your classmates.
Record any observations you make.
2. Choose your own sets of four numbers and try to build
quadrilaterals with those numbers as side lengths.
B. Use your observations from Question A.
1. Is it possible to make a quadrilateral using any four side lengths?
If not, how can you tell whether you can make a quadrilateral
from four side lengths?
2. Can you make two or more different quadrilaterals from the same
four side lengths?
3. What combinations of side lengths are needed to build rectangles?
Squares? Parallelograms?
C. 1. Use four polystrips to build a quadrilateral. Press on the sides or
corners of your quadrilateral. What happens?
2. Use another polystrip to add a diagonal connecting a pair of
opposite vertices. Now, press on the sides or corners of the
quadrilateral. What happens? Explain.
D. 1. Describe the similarities and differences between what you
learned about building triangles in Problem 4.1 and building
quadrilaterals in this problem.
2. Explain why triangles are used in building structures more often
than quadrilaterals.
Homework starts on page 76.
72
Shapes and Designs
For: Linkage Strips Activity
Visit: PHSchool.com
Web Code: amd-3402
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Mechanical engineers use the fact that quadrilaterals are not rigid to design
linkages. Below is an example of a quadrilateral linkage.
Frame
One of the sides is fixed. It is the frame. The two sides attached to the frame
are the cranks. One of the cranks is the driver and the other the follower.
The fourth side is called the coupler. Quadrilateral linkages are used in
windshield wipers, automobile jacks, reclining lawn chairs, and handcars.
In 1883, the German mathematician Franz Grashof suggested an interesting
principle for quadrilateral linkages: If the sum of the lengths of the shortest
and longest sides is less than or equal to the sum of the lengths of the
remaining two sides, then the shortest side can rotate 360°.
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Investigation 4 Building Polygons
73
TEKS I TAKS
_
6(11)( Select an appropriate problem-solving strategy including looking for a pattern.
6(13)A Make conjectures from patterns.
The Quadrilateral Game will help you explore the properties of
quadrilaterals. The game is played by two teams. To play, you need two
number cubes, a game grid, a geoboard, and a rubber band.
74
Shapes and Designs
Properties of Quadrilaterals
A. Play the Quadrilateral Game, Keep a record of interesting strategies
and difficult situations. Make notes about when you do not receive
a point during a turn. Why did you not need to move any corners
on those turns?
For: Quadrilateral Game Activity
Visit: PHSchool.com
Web Code: amd-3403
B. Write two new descriptions of quadrilaterals that you could include
in the game grid.
_
Homework starts on page 76.
Quadrilateral Game Grid
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Investigation 4 Building Polygons
75
Applications
Follow these directions for Exercises 1-4.
• If possible, build a triangle with the given set of side lengths.
Sketch your triangle.
• Tell whether your triangle is the only one that is possible. Explain.
• If a triangle is not possible, explain why.
1. Side lengths of 5, 5, and 3
2. Side lengths of 8, 8, and 8
3. Side lengths of 7,8, and 15
4. Side lengths of 5,6, and 10
5. Which set(s) of side lengths from Exercises 1-4 can make each of the
following shapes?
a. an equilateral triangle
b. an isosceles triangle
c. a scalene triangle
d. a triangle with at least two angles of the same measure
For Exercises 6 and 7, draw the polygons described to help you answer the
questions.
6. What must be true of the side lengths in order to build a triangle with
three angles measuring 60°? What kind of triangle is this?
7. What must be true of the side lengths in order to build a triangle with
only two angles the same size? What kind of triangle is this?
8. Giraldo and Maria are building a tent. They
have two 3-foot poles. In addition, they
have a 5-foot pole, a 6-foot pole, and a
7-foot pole. They want to make a
triangular-shaped doorframe for the tent
using both 3-foot poles and one of the
other poles. Which of the other poles could
be used to form the base of the door?
76
Shapes and Designs
Follow these directions for Exercises 9-12.
• If possible, build a quadrilateral with the given set of side lengths.
Sketch your quadrilateral.
• Tell whether your quadrilateral is the only one that is possible. Explain.
• If a quadrilateral is not possible, explain why.
9. Side lengths of 5, 5, 8, and 8
10. Side lengths of 5, 5, 6, and 14
11. Side lengths of 8, 8, 8, and 8
12. Side lengths of 4, 3, 5, and 14
13. Which set(s) of side lengths from Exercises 9-12 can make each of the
following shapes?
a. a square
b. a quadrilateral with all angles the same size
c. a parallelogram
d. a quadrilateral that is not a parallelogram
14. A quadrilateral with four equal sides is called a ftHgmlj~
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15. A quadrilateral with at least one pair of parallel sides is called a
Which set(s) of side lengths from Exercises 9-12 can make
a trapezoid?
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For Exercises 16 and 17, draw the polygons described to help you answer
the questions.
16. What must be true of the side lengths of a polygon to build a square?
17. What must be true of the side lengths of a polygon to build a rectangle
that is not a square?
18. Li Mei builds a quadrilateral with sides that are each five inches long.
To help stabilize the quadrilateral, she wants to insert a ten-inch
diagonal. Is this possible? Explain.
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line
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·PlISchool.com
For: Help with Exercise 18
Web Code: ame-3418
19. You are playing the Quadrilateral Game. The shape currently on the
geoboard is a square. Your team rolls the number cubes and gets
the description "A parallelogram that is not a rectangle." What is
the minimum number of vertices your team needs to move to form
a shape meeting this description?
Investigation 4 Building Polygons
20. You are playing the Quadrilateral Game. The shape currently on the
geoboard is a non-rectangular parallelogram. Your team rolls the
number cubes and gets the description "A quadrilateral with two
obtuse angles." What is the minimum number of vertices your team
needs to move to create a shape meeting this description?
Connections
21. Multiple Choice Which one of the following shaded regions is not a
representation of ~?
A.
c.
D.
78
Shapes and Designs
80m
22. a. How are all three quadrilaterals below alike?
b. How does each quadrilateral differ from the other two?
1
3
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23. In this parallelogram, find the measure of each numbered angle.
(Got::!c~~com
For: Multiple-Choice Skills
Practice
Web Code: ame-3454
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4
24. Think about your polystrip experiments
with triangles and quadrilaterals.
What explanations can you now give
for the common use of triangular
shapes in structures like bridges and
antenna towers for radio and
television?
Investigation 4 Building Polygons
Multiple Choice For Questions 25-28, choose the symmetry or symmetries
of each shape.
25. rhombus (four equal sides)
F. rotational
G. reflectional
H. both F and G
J. none
26. regular pentagon
A. rotational
B. reflectional
c. both A and B
D. none
27. square
F. rotational
G. reflectional
H. both F and G
J. none
28. a parallelogram that is not a rhombus or a rectangle
A. rotational
B. reflectional
C. both A and B
D. none
Extensions
29. In the triangle, a line has been drawn through vertex A, parallel to line
segment Be of the triangle.
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a. What is the sum of the measures of angles 1,2, and 3?
b. Explain why angle 1 has the same measure as angle 4 and why angle
3 has the same measure as angle 5.
c. How can you use the results of parts (a) and (b) to show that the
angle sum of a triangle is 180°?
80
Shapes and Designs
30. In parts (a)-(c), explore pentagons by using polystrips or by making
sketches.
a. If you choose five numbers as side lengths, can you always build a
pentagon? Explain.
b. Can you make two or more different pentagons from the same
side lengths?
31. Refer to the Did You Know? after Problem 4.2.
a. Make a model that illustrates Grashof's principle using polystrips
or paper fasteners and cardboard strips. Describe the motion of
your model.
b. How can your model be used to make a stirring mechanism?
A windshield wiper?
32. Build the figure below from polystrips. Note that the vertical sides
are all the same length, the distance from B to C equals the distance
from E to D, and the distance from B to C is twice the distance from
A toB.
a. Experiment with holding various strips fixed and moving the
other strips. In each case, tell which strips you held fixed, and
describe the motion of the other strips.
b. Fix a strip between points F and B and then try to move strip CD.
What happens? Explain why this occurs.
Investigation 4 Building Polygons
In this investigation, you experimented with building polygons by
choosing lengths for the sides and then connecting those sides to make a
polygon. These questions will help you summarize what you have learned.
Think about your answers to these questions. Discuss your ideas with
other students and your teacher. Then write a summary of your findings in
your notebook.
1. a. How can you tell whether three line segments will form a
triangle?
b. If it is possible to build one triangle, is it also possible to build a
different triangle with the same three segments? Explain.
2. a. How can you tell whether four line segments will form a
quadrilateral?
b. If it is possible to build one quadrilateral, is it also possible to build
a different quadrilateral with the same four segments? Explain.
3. Explain why triangles are useful in building structures.
What's Next?
What information about shapes can you add to your Shapes and Designs
project?
82
Shapes and Designs
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Shapes and Designs
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